Hi,
Your information is not clear. If your information on the problem you are facing is not precise, it becomes very difficult for the forum community to read your mind.
www.addondomain.yourfirstdomain.com. is the url for a
"Sub-domain" and not an "AddOn Domain".
You can cookup up anyname you want and install any amount of sub-domain you like in your hosting account.
"An
AddOn Domain" you have to purchase and register with a Registrar like "Whois" for which you have to pay an annual fee to use. You url will look like this:
http://www.addondomain.com
Once you purchase the "newdomainname" you can add it to your hosting account as a "Addon Domain" by going to your cPanel>Addon Domain and configure you new website.
As an example you can look below at my websites:
1.
www.bumiweb.com is my Main Domain
2.
www.siapamoyanganda.com is my addon domain.